D6.1 Basic Oscillations#
D6.1.1 Motivation#
Oscillations are among the most common and important phenomena in physics. They appear in countless physical systems: a swinging pendulum, a vibrating guitar string, or the alternating current in an electrical circuit. Studying oscillations gives us insight into how systems respond to restoring forces, how energy can be transferred between kinetic and potential forms, and how complex phenomena like sound and electromagnetic waves arise from simpler repeating motions. By mastering the basics of oscillations, we build the foundation for understanding waves, resonance, and many modern technologies.
D6.1.2 What are Oscillations?#
An oscillation is a repetitive back-and-forth motion of a system about an equilibrium position. The equilibrium position is the point where the system naturally rests when undisturbed. When displaced from equilibrium, a restoring force often acts on the system, pulling it back toward equilibrium. However, because of inertia, the system overshoots and continues moving to the other side, creating a cycle of motion. This push-and-pull balance between inertia and restoring forces is at the heart of oscillatory motion.
In many idealized situations, oscillations are not random or chaotic; they are characterized by repetition and predictability. A mass on a spring, for example, will trace out the same pattern over and over, provided there is little or no energy lost to friction or other dissipative effects. This predictable nature allows physicists to describe oscillations mathematically with great precision, often using sinusoidal functions such as sines and cosines. These functions capture the repeating pattern of the system’s motion over time.
It is also important to distinguish oscillations from waves. Oscillations describe the localized motion of a single system — for example, one pendulum swinging in place or one circuit oscillating in current. Waves, on the other hand, involve oscillations that spread through space and time, carrying energy with them. For instance, the oscillation of a single air molecule back and forth in its equilibrium position does not by itself create a sound. But when this oscillation is communicated to neighboring molecules, the disturbance travels as a wave, and that is what we perceive as sound. In this way, oscillations form the building blocks of wave phenomena.
D6.1.3 Basic Characteristics of Oscillations#
Amplitude (\(A\))
The maximum displacement of the system from equilibrium.Period (\(T\))
The time it takes to complete one full cycle of motion.
Unit: seconds (s).Frequency (\(f\))
The number of cycles completed per second.
Unit: cycles/second or hertz (Hz).\[ f = \frac{1}{T} \]Angular Frequency (\(\omega\))
A measure of how rapidly the system oscillates. Unit: radians/second (rad/s).Related to frequency by:
\[ \omega = 2 \pi f = \frac{2 \pi}{T} \]
D6.1.4 Example#
The plot below shows the oscillation of a block attached to a spring. The block oscillations between \(x = -1\) m and \(x = +1\) m. This maximum variation from equilibrium is known as the amplitude (A). Hence, the amplitude for this spring-mass system is \(A = 1\) m. We can see read from the plot that the period is 0.2 s, yielding a frequency of 5 Hz.
D6.1.5 Why “Angular Frequency” Makes Sense#
At first, the phrase angular frequency sounds like it belongs only to rotational motion. After all, “angular” usually means something measured in radians around a circle.
But there is a very useful way to visualize an oscillation that naturally introduces angles—without any rotation happening in the real physical system.
A geometric picture (projection idea)#
Imagine a point moving at a constant rate around a circle of radius \(A\).
At each instant, the point has an angle \(\theta\) measured from the positive horizontal axis.
Now focus only on the horizontal coordinate of that moving point. That horizontal coordinate is a 1D projection:
As the point goes around the circle, its projection onto the horizontal axis moves back and forth in a perfectly repeating way—this is exactly the shape of sinusoidal oscillation.
Why \(\omega\) appears#
If the angle increases steadily in time, we can write:
where:
\(\omega\) tells us how fast the phase angle advances in time,
\(\phi\) sets the starting phase at \(t=0\).
Substituting into the projection equation gives the standard oscillation model:
So the word angular in angular frequency refers to the fact that oscillations can be understood as a projection of uniform phase advance, where the “cycle” is naturally measured in radians.
Why radians are the natural unit#
One full oscillation corresponds to one full cycle of phase:
That is why \(\omega\) is measured in rad/s, and why
Even though the physical motion is along a line, the phase behaves like an angle that increases steadily through cycles.